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Theorem 3.1. Let Γ denote a Moore(57, 2) graph. Then for some nonnegative integers e1, e2, e3 we have e2 e3 K(Γ) ∼= (Z/2Z)1728 ⊕(Z/13Z)1519 ⊕(Z/5Z)e1 ⊕ Z/52Z ⊕ Z/53Z .  

Theorem 3.2. Let Γ be a Moore(57, 2) graph. Let e0 denote the rank of the Laplacian matrix of Γ over a field of characteristic 5. Then either ∼ 1520−e0 2 1732−e0 3 e0−3 Syl5(K(Γ)) = (Z/5Z) ⊕ Z/5 Z ⊕ Z/5 Z or   ∼ 1521−e0 2 1730−e0 3 e0−2 Syl5(K(Γ)) = (Z/5Z) ⊕ Z/5 Z ⊕ Z/5 Z .   2. Preliminaries Let Γ be a simple graph with some fixed ordering of the vertex set V (Γ). Then the adjacency matrix of Γ is a square matrix A = (ai,j) with rows and columns indexed by V (Γ), where 1, if vertex i is adjacent to vertex j, ai,j = (0, otherwise.

Let D =(di,j) be a matrix of the same dimensions as A with the of vertex i, if i = j, di,j = (0, otherwise. CRITICAL GROUP OF A MOORE(57, 2) 3

Finally, set L = D − A. The matrix L is called the Laplacian matrix of the graph Γ, and will be our primary focus. Let ZV (Γ) denote the free abelian group on the vertex set of Γ. Then the Laplacian L can be understood as describing a homomorphism: L: ZV (Γ) → ZV (Γ). We will usually use the same symbol for both the matrix and the map. The cokernel of L, coker L = ZV (Γ)/ Im(L), always has free rank equal to the number of connected components of Γ. The torsion subgroup of coker L is known as the critical group of Γ, and is denoted K(Γ). It is an interesting fact that for a connected graph Γ, the order of K(Γ) is equal to the number of spanning trees of Γ. See [3] or [9] for proofs of these basic facts and more information. One way to compute the critical group of a graph is by finding the Smith normal form of L. Recall that if M is any m × n integer matrix then one can find square, unimodular (i.e., unit determinant) matrices P and Q so that P MQ = S, where the matrix S =(si,j) satisfies:

(1) si,i divides si+1,i+1 for 1 ≤ i< min{m, n} (2) si,j = 0 for i =6 j. Then S is known as the Smith normal form of M, and it is not hard to see that ∼ coker M = Z/s1,1Z ⊕ Z/s2,2Z ⊕··· This particular decomposition of coker M is the invariant factor de- composition, and the integers si,i are known as the invariant factors of M. The prime power factors of the invariant factors of M are known as the elementary divisors of M. The concept of Smith normal form generalizes nicely when one re- places the integers with any principal ideal domain (PID), as is well known (see, for example, [6, Chap. 12]). In what follows J and I will be used to denote the all-ones matrix and the identity matrix, respectively, of the correct sizes.

3. The Critical group of a Moore(57, 2) Throughout the rest of the paper we let Γ denote a Moore(57, 2) graph. It follows easily from the definitions that Γ is strongly regular with parameters v = 3250,k = 57,λ =0,µ =1 4 DUCEY and so the adjacency matrix A must satisfy A2 = 57I +0A + 1(J − A − I) or (3.1) A2 = 56I − A + J. From this equation one can deduce [5, Chap. 9] that A has eigenvalues 7, −8, 57 with respective multiplicities 1729, 1520, 1. The degree 57 has eigenvector the all-one vector 1; the other eigenvalues are the restricted eigenvalues. Since the graph is regular, we immediately get the Laplacian spec- trum: eigenvalues 50, 65, 0 with multiplicities as above. Kirchhoff’s Matrix-Tree Theorem [5, Prop. 1.3.4] tells us that the number of span- ning trees of Γ is the product of the non-zero eigenvalues, divided by the number of vertices. We thus get the order of the critical group of Γ: 1 |K(Γ)| = · 501729 · 651520 3250 =21728 · 54975 · 131519. We remark that the number of such abelian groups is quite large. The next theorem begins to narrow things down. Let Sylp(K(Γ)) denote the Sylow p-subgroup of the critical group. Theorem 3.1. Let Γ denote a Moore(57, 2) graph. Then for some nonnegative integers e1, e2, e3 we have e2 e3 K(Γ) ∼= (Z/2Z)1728 ⊕(Z/13Z)1519 ⊕(Z/5Z)e1 ⊕ Z/52Z ⊕ Z/53Z . Proof.   Substituting A = 57I − L into equation 3.1, we get (57I − L)2 = 56I − (57I − L)+ J L2 − 115L = −3250I + J (3.2) (L − 115I)L = −(2 · 53 · 13)I + J. This last equation tells us much about the Smith normal form of L. As in the previous section, we view L as defining a homomorphism of free Z-modules L: ZV (Γ) → ZV (Γ). Define a subgroup of ZV (Γ):

Y = a v a =0 .  v v  v∈V (Γ) v∈V (Γ)  X X

  CRITICAL GROUP OF A MOORE(57, 2) 5

Note that Y is the smallest direct summand of ZV (Γ) that contains Im L (i.e., it is the purification of Im L). Changing the codomain of L to Y does not affect the nonzero invariant factors of L, so we do. In fact, with this adjustment we have coker L ∼= K(Γ). If we also restrict the domain of L to Y the Smith normal form will probably be altered. However, note that coker L is a quotient of coker L|Y . As Y = ker J, from equation 3.2 we get 3 (3.3) (L − 115I)|Y L|Y = −(2 · 5 · 13)I.

Take any pair of integer bases for Y which put the matrix for L|Y into Smith normal form. Follow a basis element x through the composition of maps on the left side of equation 3.3; we can see that the image is 3 −(2 · 5 · 13)x. Hence the invariant factor of L|Y associated to the basis element x must divide 2·53 ·13. Said another way, the elementary divi- 2 3 sors of L|Y can only be from among {2, 13, 5, 5 , 5 }, and so coker L|Y has a cyclic decomposition of the form in the statement of the theorem. The same must be true for its quotient K(Γ).  Remark. A bicycle of Γ is a subgraph for which every vertex has even degree and whose edges form an edge-cutset of Γ (i.e., the deletion of the edges in the subgraph results in Γ becoming disconnected). The set of all bicycles of Γ form a binary vector space with operation symmetric difference of edges. The dimension of this vector space is equal to the number of invariant factors of L that are even [7, Lem. 14.15.3]. Thus we have shown that Γ has 21728 bicycles–the maximum possible for the order of its critical group. In the next theorem we will flesh out a relationship between the integers e1, e2, e3 and the 5-rank of L, which we denote by e0. As Syl5(K(Γ)) is the mystery here, it will be convenient to ignore all other primes than 5. We now briefly explain how to do this. For a prime integer p, let Zp denote the ring of p-adic integers. The ring Zp is a PID, so Smith normal form still makes sense for matrices with entries from Zp; this of course encompasses all integer matrices. When we view an integer matrix as having entries from the ring Zp, the elementary divisors that survive the change of viewpoint are the powers of p. The elementary divisor multiplicities can then be understood in terms of certain Zp-modules attached to the matrix or map under consideration. n m Let η : Zp → Zp be a homomorphism of free Zp-modules of finite rank. We get a descending chain of submodules of the domain n Zp = M0 ⊇ M1 ⊇ M2 ⊇··· 6 DUCEY by defining n i m Mi = x ∈ Zp | η(x) ∈ p Zp . That is, M consists of the domain elements whose images under η are i  divisible by pi. In a similar way, we can define −i Ni = p η(x) | x ∈ Mi . This gives us an ascending chain of modules in the codomain

N0 ⊆ N1 ⊆ N2 ⊆··· m that will eventually stabilize to the purification of Im η in Zp . For a ℓ submodule R of the free Zp-module Zp , we define ℓ ℓ R = R + pZp /pZp .

Note that R is a vector space over the finite field Fp = Zp/pZp. We denote the field of fractions of Zp by Qp. n m Lemma 3.1. Let η : Zp → Zp be a homomorphism of free Zp- i modules of finite rank. Let ei denote the multiplicity of p as an el- ementary divisor of η. Then, for i ≥ 0,

dimFp Mi = dimFp ker(η)+ ei + ei+1 + ··· and

dimFp Ni = e0 + e1 + ··· + ei. Proof. Take a basis B of the domain and a basis C of the codomain for which the matrix of η is in Smith normal form. For i ≥ 0, define the subset of B i i+1 Bi = {x ∈ B | p divides η(x), but p ∤ η(x)}.

Then the basis B is partitioned by the sets {Bi} along with D = {x ∈ B | η(x)=0}. In other words, we split B up so that basis elements associated to the same invariant factor are grouped together. Note that Bi has cardinal- ity ei and D is a basis for ker(η). A little thought reveals that a basis for Mi is given by the set

i i−1 D ∪ p B0 ∪ p B1 ∪···∪ pBi−1 ∪ Bk . k≥i ! [ The nonzero elements of the Fp-reduction of this set yields a basis of Mi, and the first part of the lemma is proved. By considering a similar partition of C the second part of the lemma becomes clear as well.  CRITICAL GROUP OF A MOORE(57, 2) 7

Theorem 3.2. Let Γ be a Moore(57, 2) graph. Let e0 denote the rank of the Laplacian matrix of Γ over a field of characteristic 5. Then either ∼ 1520−e0 2 1732−e0 3 e0−3 Syl5(K(Γ)) = (Z/5Z) ⊕ Z/5 Z ⊕ Z/5 Z or   ∼ 1521−e0 2 1730−e0 3 e0−2 Syl5(K(Γ)) = (Z/5Z) ⊕ Z/5 Z ⊕ Z/5 Z . Proof.   We view the Laplacian matrix L of Γ as a matrix over Z5. For λ an eigenvalue of L, let Vλ denote the Q5-eigenspace for λ. One sees that V (Γ) V (Γ) V (Γ) V65 ∩ Z5 ⊆ N1, and so V65 ∩ Z5 ⊆ N1. Since V65 ∩ Z5 is a direct V (Γ) summand of Z5 (being the kernel of the endomorphism L − 65I of V (Γ) the Z5-lattice Z5 ) with rank equal to the dimension of V65 over Q5, V (Γ) we have that dimQ5 V65 = dimF5 V65 ∩ Z5 . Applying Lemma 3.1,

(3.4) 1520 = dimQ5 V65 V (Γ) = dimF5 V65 ∩ Z5

≤ dimF5 N1

= e0 + e1.

V (Γ) By a similar argument, V50 ∩ Z5 ⊆ M2 and Lemma 3.1 implies that

(3.5) 1729 = dimQ5 V50 V (Γ) = dimF5 V50 ∩ Z5

≤ dimF5 M2

=1+ e2 + e3. Note that ker L is spanned by the all-one vector 1, which explains the 1 appearing in the right hand side of the above inequality. Now consider carefully these two inequalities 3.4 and 3.5:

1520 ≤ e0 + e1

1729 ≤ 1+ e2 + e3. The sum of the left hand sides is 1520+1729 = 3249, while the sum of the right hand sides is e0 + e1 + e2 + e3 +1 = 3250. There are exactly two ways in which this can be:

Case 1: 1520 = e0 + e1 and 1729 = e2 + e3.

Case 2: 1521 = e0 + e1 and 1728 = e2 + e3. 8 DUCEY

There is another equation that applies to all cases. Since

4975 |Syl5(K(Γ))| =5 , we have

(3.6) 4975 = e1 +2e2 +3e3. Taking equation 3.6 with the two equations of Case 1, we are seeking nonnegative integer solutions to the system

e0 + e1 = 1520

e2 + e3 = 1729

e1 +2e2 +3e3 = 4975.

This is easily done by hand. Choosing, say, e3 to be free we get:

• e3 = t • e2 = 1729 − t • e1 = 1517 − t • e0 =3+ t.

Writing each unknown in terms of the 5-rank e0 instead gives us the first isomorphism in the statement of the theorem. In Case 2, the system becomes

e0 + e1 = 1521

e2 + e3 = 1728

e1 +2e2 +3e3 = 4975. The solutions may be written

• e3 = t • e2 = 1728 − t • e1 = 1519 − t • e0 =2+ t.

If we instead take e0 to be free we get multiplicities as in the second isomorphism of the theorem. 

Remark. The author has thus far been unable to obtain strong bounds on the possible 5-rank of L. The ambitious reader is directed to [4]; there the authors compute the relevant p-ranks of the and the Hoffman-Singleton graph. Knowledge of specific adjacencies and constructions within the graphs are used. CRITICAL GROUP OF A MOORE(57, 2) 9

4. Acknowledgements The author thanks an anonymous referee for helpful comments. This work was supported by James Madison University’s Tickle Fund. References [1] James Alexander and Tim Mink. A new method for enumerating independent sets of a fixed size in general graphs. J. , 81(1):57–72, 2016. [2] Michael Aschbacher. The nonexistence of rank three permutation groups of degree 3250 and subdegree 57. J. Algebra, 19:538–540, 1971. [3] N. L. Biggs. Chip-firing and the critical group of a graph. J. Algebraic Combin., 9(1):25–45, 1999. [4] A. E. Brouwer and C. A. van Eijl. On the p-rank of the adjacency matrices of strongly regular graphs. J. Algebraic Combin., 1(4):329–346, 1992. [5] Andries E. Brouwer and Willem H. Haemers. Spectra of graphs. Universitext. Springer, New York, 2012. [6] David S. Dummit and Richard M. Foote. Abstract algebra. John Wiley & Sons, Inc., Hoboken, NJ, third edition, 2004. [7] Chris Godsil and Gordon Royle. Algebraic graph theory, volume 207 of Grad- uate Texts in Mathematics. Springer-Verlag, New York, 2001. [8] A. J. Hoffman and R. R. Singleton. On Moore graphs with diameters 2 and 3. IBM J. Res. Develop., 4:497–504, 1960. [9] Dino J. Lorenzini. A finite group attached to the Laplacian of a graph. Discrete Math., 91(3):277–282, 1991. [10] Martin Maˇcaj and Jozef Sir´aˇn.ˇ Search for properties of the missing Moore graph. Linear Algebra Appl., 432(9):2381–2398, 2010.

Dept. of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807 E-mail address: [email protected]